This post describes the Sysudoku basic technique of line marking, which pencil marks all remaining candidates, line slinks and bi-value cells. In a walk through line marking of the Sue de Coq classic, the efficiencies of line marking are explained.
The Sue de Coq post puzzle provides a good introduction to the mechanics of line marking. To the box marked grill, we add green borders cells known to be bv (bi-value) cells, the ones with two candidates.
Without all candidates in place, how do we know which cells are bv? Naked pairs like Nnp39 and W14 are obvious. Two cells left in their lines, and two missing numbers to fill them. Room for nothing else. Alignment is also the key to the others. In chute Wc5, 3, 6 and 7 must fit in three cells, and the placements are determined. Same story in chute Ec5.
Easier puzzles are barely alive after box marking, and will collapse upon our filling only a few lines. Harder puzzles will push us to fill more lines, perhaps all of them. Therefore, it’s important to know the most efficient way to order the lines to be filled. Least effort first, lazy bones! The greedy strategy. The actual easiest lines are never known, but the number of free cells is a good predictor of effort to fill. So we order the lines to fill by increasing free cells.
Each marking can change the remaining order of lines, so we don’t actually order the lines, but just select the next line with the fewest free cells to fill. There will be ties, so we scan down rows and across columns looking for the first line with as many free cells as the last line filled. After it’s filled, we continue from there. When there are no more, we start over with the next larger number of free cells.
Actually, we don’t count free cells. We count the number of locked cells, because it’s easier. What are locked cells? When n numbers in a unit are confined to n cells of the unit, no more numbers can be added. We say that the sets of cells and candidates are locked. The cells of the unit not locked are the free cells. Clues are locked. There are no free cells in row 5 above. There are 5 free cells in row 6.
Do you have your copy, updated as needed, to the grid above? If so, we can walk on through line marking with it.
Here we discover two lines with only three free cells, r9 and c9. We mark rows before columns, so our line marking trace begins with
In line marking, we fill the cells of the line together, taking advantage of what they have in common, the unplaced missing numbers. In r9 the missing numbers are 1, 5, and 9. We have placed the 9 candidates, and have to determine where 1 and 5 can go. I make up a fill string of unplaced, missing numbers and place it at the end of the line.
Then for each cell, I match the fill string against the locked numbers along the crossing line and box of the cell, deleting. In ©PowerPoint, I actually place a copy of the fill string in the cell and delete matching digits from it, then leave what is left in the cell. This ritual is very efficient as fill strings can get longer. On this line, I added 15 to r9c1, 1 to r9c2 and nothing to r9c8. This left me with two more bv to mark.
On the other 3f:, c9, I see 28 for the fill string already, plus the placed 5 makes 3, accounting for the three free cells.
Now the 3f:’s are gone. How about working on your updated copy and go through the 4f: rows before checking with the diagram below.
Ready? Let’s go over it line by line:
On r9, I’m sure you have your bv marked, but you may have your 5r9c1 in a different spot. The lower left hand corner placement marks the line slink. I deleted the 5 and dragged a copy over from r9c8. When the line is filled, check each fill string digit for line slinks and adjust the position marks.
Next is r1, where 5 is left off the fill string because they are already place. That’s always the case when the candidates are clues or box slinks. The 5 line slink is there, but it’s not necessary to mark it. That’s for later, but it’s covered for now in that box marks are higher priority than line marks.
For r3, I copied the 389 string from r1, deleted the 8, then added 2. That’s another advantage of ©PowerPoint. Not 4, already placed. Note that the string length should be the number of free cells, less the number of placed numbers. You don’t want to leave out a number. Did you delete the numbers locked in boxes?
As you skip on down now, do you agree that marking filled lines like r5 and r6 (—) helps? For r8, I copied the r9 fill string, and deleted 1, adding 2. Strings over the same boxes tend to share digits.
Continuing with the 5f: lines, what is your fill string for r7? If no clues are added, would you rather do one 6f: row, or five columns? Me too. So finish your copy and check it below where we finish line marking with something called closure.
When all lines in one direction are marked, every cell is filled with pencil marks for remaining candidates, and all bv should be marked. But there are follow up actions that have not been performed for some lines in the other direction, the lines that were not assigned a fill string. Closure is the follow up actions performed on these lines.
One of the follow up actions is the marking of line slinks. Here is the grid of Sue de Coq’s classic puzzle, as row marking is completed.
Starting with c1, we close the columns, by scanning each one for slinks. The full closure, with follow up on subsets and X-wings, is described later. To close a line, account for each number. Numbers 1, 3 and 5 have three candidates in c1. Numbers 2, 7 and 8 are represented by clues. Number 6 has a slink, but it’s row slinks have marking priority. The candidate 9r2c1 should move to the right corner, to mark the column slink.
Now it’s up to you to complete the slink marking closure on the remaining columns, and check your result with the line marked grid of the slink marking post of 10/04/11.
Here is a striking example of naked and hidden singles. It’s a Friday four star from Dave Green, who composes the puzzles in my newspaper, the Akron Beacon Journal. Do the box marking on this one, and you will be primed for a line marking walkthrough with me. It came along, on December 6, 2013.
Here is a striking example of naked and hidden singles. It’s a Friday four star from Dave Green, who composes the puzzles in my newspaper, the Akron Beacon Journal. Do the box marking on this one, and you will be primed for a line marking walkthrough with me. It came along, on December 6, 2013